Surface plasmons on a randomly rough surface

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Abstract
We have studied the dispersion relation for surface plasmons on a randomly rough surface, going beyond the lowest-order approximation in the surface profile function. This improvement is required because an analysis of the expansion for the surface-plasmon dispersion relation in powers of the surface profile function shows that it contains an infinite subset of terms that are all of the same order of magnitude as the lowest-order contribution, which is the only one that has been considered in previous theoretical determinations of this dispersion relation. The indicated subset of terms is summed to yield a nonlinear integral equation for the surface-plasmon proper self-energy, in terms of which the dispersion relation is expressed. This integral equation has been solved numerically and the surface-plasmon spectral density constructed. The splitting of the surface-plasmon dispersion curve into two branches by the surface roughness, which was predicted theoretically by the lowest-order perturbation theory calculation, and which has been observed experimentally, is preserved in the results of the present calculation. However, both the magnitude of the splitting and the damping of the surface plasmon obtained from the present calculations are larger, for the same corrugation strength, than the same quantities obtained from the lowest-order perturbation calculation, for most surface-plasmon wave vectors.